Dynamic Models of Reputation and Competition in Job-Market Matching
aa r X i v : . [ c s . G T ] D ec Dynamic Models of Reputation and Competition in Job-MarketMatching
Jon Kleinberg ∗ Sigal Oren † Abstract
A fundamental decision faced by a firm hiring employees — and a familiar one to anyone whohas dealt with the academic job market, for example — is deciding what caliber of candidatesto pursue. Should the firm try to increase its reputation by making offers to higher-qualitycandidates, despite the risk that the candidates might reject the offers and leave the firm empty-handed? Or is it better to play it safe and go for weaker candidates who are more likely to acceptthe offer? The question acquires an added level of complexity once we take into account theeffect one hiring cycle has on the next: hiring better employees in the current cycle increasesthe firm’s reputation, which in turn increases its attractiveness for higher-quality candidates inthe next hiring cycle. These considerations introduce an interesting temporal dynamic aspectto the rich line of research on matching models for job markets, in which long-range planningand evolving reputational effects enter into the strategic decisions made by competing firms.The full set of ingredients in such recruiting decisions is complex, and this has made it difficultto model the fundamental strategic tension at the core of the problem. Here we develop a modelbased on two competing firms to try capturing as cleanly as possible the elements that we believeconstitute this strategic tension: the trade-off between short-term recruiting success and long-range reputation-building; the inefficiency that results from underemployment of people who arenot ranked highest; and the influence of earlier accidental outcomes on long-term reputations.Our model exhibits all these phenomena in a stylized setting, governed by a parameter q that captures the difference in strength between the top candidate in each hiring cycle and thenext best. Building on an economic model of competition between parties of unequal strength,we show that when q is relatively low, the efficiency of the job market is improved by long-rangereputational effects, but when q is relatively high, taking future reputations into account cansometimes reduce the efficiency. While this trade-off arises naturally in the model, the multi-period nature of the strategic reasoning it induces adds new sources of complexity, and ouranalysis reveals interesting connections between competition with evolving reputations and thedynamics of urn processes. ∗ Cornell University, Ithaca NY. Email: [email protected]. † Microsoft Research and Hebrew University, Israel. Email: [email protected].
Introduction
Markets for employment have been the subject of several large bodies of research, including the longand celebrated line of work on bipartite matching of employers to job applicants [25], sociologicaland economic approaches to the process of finding a job [12, 19, 24], and many other frameworks.Recent work in theoretical computer science has modeled issues such as the competition amongemployers for applicants [15, 16] and hiring policies that take a firm’s reputation into account [5].Despite this history of research, there remain a number of fundamental issues in job-marketmatching that have gone largely unmodeled. One of these, familiar to anyone who has dealt withjob markets in academia or related professions, is the feedback loop over multiple hiring cyclesbetween the job candidates that a firm (or academic department) pursues and the evolution of itsoverall reputation.There is of course a very broad set of ingredients that go into the competition for job candidatesover multiple hiring cycles. This makes it challenging to abstract the basic issues into a model forthis type of multi-period competition. In the present paper we pare down this complexity to tryformulating a model that captures, as cleanly as possible, what we view to be the basic sources ofstrategic tension in the process.We develop a model based on two firms that compete for candidates over multiple periods, witha pool of candidates that has the same structure in each period; the outcome of the competition fora given candidate is determined probabilistically, based on the relative reputations of the two firmsat the time they compete. This is a highly reduced and stylized model, but it produces a complexset of behaviors that we believe should be components of any of a range of richer extensions ofthe model as well. In particular, the process of job-market competition in our model exhibits thefollowing fundamental trade-offs:(i) Successfully recruiting higher-quality candidates can raise a firm’s reputation, which in turncan make it more attractive to candidates in future hiring cycles.(ii) On the other hand, competing for these higher-quality candidates comes with a greater riskof emerging from a given hiring cycle empty-handed.(iii) The incentive to compete for top-ranked candidates can lead to underemployment of lower-ranked candidates, as they are at risk of receiving no offers while firms instead compete fortheir higher-ranked counterparts.(iv) The trajectory of the process can be heavily influenced by a small number of “accidental”recruiting outcomes in the early stages, as reputations are first being established.The trade-off between (i) and (ii) above arises from the equilibrium in the dynamic matchinggame played by the two firms with respect to the pool of available candidates. We find that inequilibrium, there is an initial period of competition, which can end when one firm decides it isso far behind the other that it is no longer worth competing for the top-ranked candidates. Forcertain natural ranges of parameters, there is in fact an interesting bifurcation — depending onthe random outcomes of the initial stages, there is a positive probability one firm will “give up” oncompeting for the best candidates, but also a positive probability that the two firms will competefor the best candidates forever.The issue in point (iii) is a question of efficiency: if a firm’s utility is the total quality of allthe candidates it hires, then our measure of social welfare — the sum of the firms’ utilities —is simply the total quality of all candidates hired by either of them. We can then consider the1atural performance guarantee question in this model: how does the social welfare under multi-period strategic behavior compare to the maximum social welfare attainable, where the maximumcorresponds to a central authority that is able to impose a matching of candidates to firms? Weobtain a tight bound of 21 + √ . ≈ .
898 on the ratio of the social welfare under the canonicalNash equilibrium to the optimal social welfare in this model, as the number of periods goes toinfinity. The exact numerical bound here will of course be a property of our modeling decisions,but the trade-off that leads to it seems inherent in the structure of the multi-period competition.Moreover, studying this performance ratio as a function of the number of periods, we find thatfor some settings of the parameters, the performance ratio is worse for instances with a “medium”number of periods, rather than those with very few (where long-range planning does not haveenough force to favor competition over star candidates) or those with very many (where the weakerfirm is likely to stop competing, leading to a higher level of overall employment).Finally, the issue in point (iv) — the “accidental” effects of early competition outcomes —turns out to be analyzable in our model via a concrete connection to Polya urn processes [22]. Weshow how the evolution of the two firms’ reputations can be tracked through an analysis that isclosely related to the evolving composition of a Polya urn; however, the analysis is made morecomplicated by the fact that the steps in the process are under the control of strategic agents whoare calculating their actions inductively with respect to the expected outcomes in future periods.This particular combination of phenomena (i)-(iv) appears to be new from an analytical per-spective using formal models, despite its familiarity from everyday experience, and its connectionswith strands of more empirical and ethnographic work in economics [29] and sociology [7, 23]. Wethus view the reduced-form model developed here as correspondingly shedding light on the interplaybetween the inherent strategic and probabilistic considerations as the process unfolds — includingthe emergence from the model of qualitative principles such as the transition between long-runningcompetition and a decision by one firm to “give up” and accept a lower rank. Moreover, the use ofa model with two firms is consistent with the long-standing style of analysis in terms of duopoliesfor multi-period game-theoretic models (see e.g. [2, 3, 17, 21]); two-firm competition is often theinitial place where one looks for principles in establishing such models.As a last point, we note that while our model is expressed in terms of job-market recruiting,there are many settings in which firms compete over multiple time periods, making decisions thathave effects on their reputations and hence their relative performance in the future. As such, thetype of probabilistic analysis we carry out here for the underlying dynamic multi-period processmay be useful in thinking about the strategic management of evolving reputations more generally— thought still of course in the highly reduced form in which we have expressed it. For example,it would be interesting to see whether our framework can be adapted to settings where relatedissues have been explored, including the study of product compatibility [6]. The issue of whethera weaker firm decides to directly compete with a stronger one, or to avoid direct competition, isalso implicit in studies of the branding and advertising decisions firms make — including whetherto explicitly acknowledge a second-place status, such as the example (discussed in [20]) of the Aviscar rental company’s “We Try Harder” campaign. Formulating the model
We now describe the model and its underlying parameters in moredetail. Again, we stress that our model is designed to produce the essential phenomena in thismulti-period competition as cleanly as possible, and hence is built on two firms that compete in a We use the neutral term “performance ratio” rather than price of anarchy or price of stability because — as wewill see — our game has a natural equilibrium, and we are interested in the relative performance of this naturalequilibrium, rather than necessarily focusing on the best or worst equilibrium. k rounds. We can think of each playeras representing an academic department that is able to try hiring one new faculty candidate ineach of the next k hiring seasons. In each round t ∈ { , , . . . , k } , the players are presented witha set of job candidates with fixed numerical qualities . Since we have only two firms in our model,we will assume that the firms’ hiring will only involve considering the two strongest candidates;we therefore assume that there are only two candidates available. Normalizing the quality of thestronger candidate, we define the qualities of the two candidates to be 1 and q < utility — the total quality of allcandidates it has hired — and its reputation — its ability to attract new candidates based on thequality of the people it has hired. A number of studies of academic rankings have emphasized thatdepartments are judged in large part by their strongest members; intuitively, this is why a smallerdepartment with several “star” members can easily rank higher than a much larger department,and ranking schemas often include measures that focus on this distinction.Given this, a natural way to define reputation in our model is to say that the reputation offirm i in round t , denoted x i ( t ), is equal to the number of higher-quality candidates (i.e. those ofquality 1 rather than q ) that it has hired so far. This is distinct from the utility of firm i in round t , denoted u i ( t ), which is simply the sum of the qualities of all the candidates it has hired.We assume that a firm is seeking to maximize its utility over the full k rounds; however ,note that since this is a multi-period game, and reputation determines success in future roundsof hiring, a firm’s equilibrium strategy will in fact involve actions that are effectively seeking toincrease reputation even at the expense of short-term sacrifices to expected utility. This, indeed,is exactly the type of behavior we hope to see in a model of recruiting.Building on this discussion, we therefore structure the game as follows. • Each player i has a numerical reputation x i ( t ) and utility u i ( t ) in round t . We will focusmainly on the case in which the two players each start with reputation equal to 1, though inplaces we will consider variations on this initial condition. • In each round t ∈ { , , . . . , k } , player i chooses one of the candidates j to try recruiting; thischoice of j constitutes the player’s strategy in round t . • If player i is the only one to try recruiting j , then j accepts the offer. If both players competefor the same candidate j , then j accepts player i ’s offer with probability proportional to player i ’s reputation. This follows the Tullock contest function that is standard in economic theoryfor modeling competition [27, 28], thus we have : player 1 hires j with probability x ( t ) x ( t ) + x ( t ) and player 2 hires j with probability x ( t ) x ( t ) + x ( t ) . The player who loses thiscompetition for candidate j hires no one in this round. • Finally, each player receives a payoff in round t equal to the quality of the candidate hiredin the round (if any). The player’s utility is increased by the quality of the candidate it hashired; the player’s reputation is increased by 1 if it has hired the stronger candidate in round t , and remains the same otherwise.Thus the model captures the basic trade-off inherent in recruiting over multiple rounds — bycompeting for a stronger candidate, a player has the opportunity to increase its reputation by alarger amount, but it also risks hiring no one. The model is designed to arrive at this trade-off3sing very few underlying parameters; but we believe that the techniques developed for the analysissuggest approaches to more complex variants, and we discuss some of these in the conclusions section(Section 5).The maximum possible social welfare is achieved if the two players hire the top two candidatesrespectively in each round, achieving a social welfare of k (1 + q ). The key question we consider hereis what social welfare can be achieved in equilibrium for this k -round game, and how it comparesto the welfare of the social optimum. In effect, how much does the struggle for reputation leavecandidates unemployed?The subgame perfect equilibria in this multi-round game are determined by backward induction— essentially, in a given round t , a player evaluates the possible values its utility and reputationcan take in round t + 1, after the (potentially probabilistic) outcome of its recruiting in round t . There are multiple equilibria, but there is a single natural class of canonical equilibria for themodel, in which the higher-reputation player always goes after the stronger candidate, and —predicated on the equilibrium having this form in future rounds — the lower-reputation playermakes an optimal decision to either compete for the stronger candidate or make an offer to theweaker candidate. (When the lower-reputation player is indifferent between these two options,we break the symmetry using the assumption that the lower player hires the weaker candidate.)The canonical equilibrium can be also viewed as the result of a best response order in which atevery round the higher-reputation gets the advantage of making the first choice. Proving thatthis structure in fact produces an equilibrium is non-trivial; in part this is because reasoningabout subgame perfect equilibria always involves some complexity due to the underlying tree ofpossibilities, but the present model adds to this complexity because the randomization involved inthe outcome causes the possible trajectories of the game to “explore” most of this tree.We study the behavior of this canonical equilibrium, and we define the performance ratio of aninstance to be the ratio of total welfare between the canonical equilibrium and the social optimum. Overview of Results
We first consider the performance ratio as a function of the number ofrounds k . As an initial question, which choice of k yields the worst performance ratio? When q < , the answer is simple: for k = 1, the players necessarily compete in the one round theyhave available, and this yields a performance ratio of 1 / (1 + q ) — as small as possible. When q > , however, the situation becomes more subtle. For k = 1, the players do not compete inthe canonical equilibrium, and so the performance ratio for k = 1 is 1. At the other end of thespectrum, when q ≥ , the two players will eventually stop competing with probability 1 and theperformance ratio converges up to 1 when k becomes large. But in between, the performance ratiocan be larger than at both extremes; in particular, when the quantity q − q approaches an integervalue k from below, we show that the performance ratio is maximized when the number of roundstakes this intermediate value k . These results show how the time scale over which the players takereputational effects into account can have a subtle (and in this case non-monotonic) effect on theefficiency of the job market.We then turn to the main result of the paper, which is to analyze the performance ratio in thelimit as the number of rounds k goes to infinity. When q ≥ , as just noted, we show that the twoplayers will eventually stop competing with probability 1 and the performance ratio converges to 1.But when q < , something more complex happens: there is a positive probability, strictly between0 and 1, that the players compete forever. This has a natural interpretation — as reputationsevolve, the two players can settle into relative levels of reputation under which it is worthwhile forthe lower player to compete for the stronger candidate; but it may also happen that after a finitenumber of rounds, one player decides that it is too weak to continue competing for the stronger4andidate, and it begins to act on its second-tier status. What is interesting is that each of theseoutcomes has a positive probability of occurring.The possibility of indefinite competition leads to a non-trivial performance ratio; we show thatthe worst case occurs when q = √ . − ≈ . √ . ≈ . q goes either to 0 or to 1. Our analysisproceeds by defining an urn process that tracks the evolution of the players’ reputations; this is anatural connection to develop, since urn processes are based on models in which probabilities ofoutcomes in a given step — the result of draws from an urn — are affected by the realized outcomesof draws in earlier steps. We provide more background about urn processes in the next section.Informally speaking, the fact that a player might compete for a while and then permanently give upin favor of an alternative option is also reminiscent of strategies in the multi-armed bandit problem,where an agent may experiment with a risky option for a while before permanently giving up andusing a safer option; later in the paper, we make this analogy more precise as well. To make useof these connections, we study a sequence of games that begins with players who are constrainedto follow a set sequence of decisions for a long prefix of rounds, and we then successively relax thisconstraint until we end up with the original game in which players are allowed to make strategicdecisions from the very beginning.In Appendix C, we also consider variants of the model in which one changes the function usedfor the success probabilities in the competition between the two players for a candidate. Notethat the way in which competition is handled is an implicit reflection of the way candidates formpreferences over firms based on their reputations, and hence varying this aspect of the model allowsus to explore different ways in which candidates can behave in this dimension. In particular, weconsider a variation on the model in which — when the two players compete for a candidate — thelower-reputation player succeeds with a fixed probability p < and the higher-reputation playersucceeds with probability 1 − p . This model thus captures the long-range competition to becomethe higher-reputation player using an extremely simple model of competition within each round.The main result here is that for all p < q , the performance ratio converges to 1; the analysis makesuse of biased random walks in place of urn processes to analyze the long-term competition betweenthe players. Further Related Work
As noted above, there has been recent theoretical work studying theeffect of reputation and competition in job markets. Broder et al. consider hiring strategies designedto increase the average quality of a firm’s employees [5]. Our focus here is different, due to thefeedback effects from future rounds that our model of competition generates: a few weak initialhires can make it very difficult for a player to raise its quality later, while a few strong initial hirescan make the process correspondingly much easier. Immorlica et al. consider competition betweenemployers, though in a quite different model where candidates are presented one at a time as inthe secretary problem [15, 16], and each player’s goal is to hire a candidate that is stronger thanthe competitor’s. They do not incorporate the spillover of this competition into future rounds.Our work can also be viewed as developing techniques for analyzing the performance ratioand/or price of anarchy in settings that involve dynamic matchings — when nodes on one side ofa bipartite graph must make strategic decisions about matchings to nodes that arrive dynamicallyto the other side of the graph. In the context of job matching, Shimer and Smith consider adynamic matching model of a labor market in which the central constraint is the cost of searchingfor potential partners [26]. Haeringer and Wooders apply dynamic matching to the problem ofsequential job offers over time [13], but in a setting that considers the sequencing of offers in asingle hiring cycle; this leads to different questions, since the consequence for reputation in future5iring cycles is not in the scope of their investigation. Dynamic matchings have also been appearingin a number of other recent application contexts (e.g. [8, 30]), and there are clearly many unresolvedquestions here about the cost of strategic behavior.
An instance of the recruiting game, as described in the introduction, is defined by the initialreputations x and x of the two players; the relative quality q of the weaker candidate comparedto the stronger one; and the number of rounds k . Accordingly, we denote an instance of the gameby G k,q ( x , x ). Generally q will be clear from context, and so we will also refer to this game assimply G k ( x , x ). We will refer to the player of higher reputation as the higher player , and theplayer of lower reputation as the lower player . In case the players have the same reputation we willrefer to player 1 as the higher player.The game as defined is an extensive-form game, and as such it can admit many subgame perfectequilibria. For example, it is easy to construct a single-round game in which it is an equilibriumfor the lower player to try to recruit the stronger candidate and for the higher player to go afterthe weaker candidate. This equilibrium clearly has a less natural structure than one in which thehigher player goes after the stronger option; to avoid such pathologies, as noted in the introduction,we will study multi-round strategies s k ( x , x ) that are defined as follows: Definition 2.1
Denote by s k ( x , x ) the following strategies for the players over the k rounds: inevery round the higher player goes for the stronger candidate and the lower player best-respondsby choosing the candidate that maximizes its utility, taking into account the current round and alllater rounds by induction.For s k ( x , x ) to be well-defined we make the following two assumptions: (1) If the lower playeris indifferent between going for the stronger candidate and the weaker candidate we assume itchooses to go for the weaker candidate. (2) If the two players have the same reputations we breakties in favor of player . The strategies s k ( x , x ) can be summarized essentially by saying that in every round of thegame, first the higher player gets to make an offer to its preferred candidate, and given this decisionthe lower player makes the choice maximizing its utility. To show that the strategies s k ( x , x )form a sub-game perfect equilibrium we will show inductively that in every round it is in thehigher player’s best interest to make an offer to the stronger candidate. More formally we denotethe strategy of making an offer to the stronger candidate in some round by + and to the weakercandidate by − . We define f ( s k ( x , x )) to be the pair of strategies that the players use in the firstround of s k ( x , x ).We denote player i ’s utility when the two players play the strategies prescribed by s k ( x , x )by u i ( s k ( x , x )). We now formally write down the utility of the players in s k ( x , x ) based on thevalue of f ( s k ( x , x )): • If f ( s k ( x , x )) = h + , + i then u ( s k ( x , x )) = x x + x (1 + u ( s k − ( x + 1 , x ))) + x x + x u ( s k − ( x , x + 1)) u ( s k ( x , x )) = x x + x u ( s k − ( x + 1 , x )) + x x + x (1 + u ( s k − ( x , x + 1))) . If f ( s k ( x , x )) = h + , −i then u ( s k ( x , x )) = 1 + u ( s k − ( x + 1 , x )) u ( s k ( x , x )) = q + u ( s k − ( x + 1 , x )) • If f ( s k ( x , x )) = h− , + i then u ( s k ( x , x ) = q + u ( s k − ( x , x + 1)) u ( s k ( x , x )) = 1 + u ( s k − ( x , x + 1))We denote the social welfare of playing the strategies s k ( x , x ) by u ( s k ( x , x )) = u ( s k ( x , x )) + u ( s k ( x , x )) . Even though it is natural to suspect that the strategies s k ( x , x ) are indeed a sub-game perfectequilibrium, proving that this is the case is not such a simple task. The first step in showingthat the strategies s k ( x , x ) are a sub-game perfect equilibrium, and a useful fact by itself, is themonotonicity of the players’ utilities u i ( s k ( x , x )). More formally, in Section A of the appendixwe show that: Claim 2.2
For any x , x , and ǫ > :1. u ( s k ( x + ǫ, x )) ≥ u ( s k ( x , x )) ≥ u ( s k ( x , x + ǫ )) .2. u ( s k ( x , x + ǫ )) ≥ u ( s k ( x , x )) ≥ u ( s k ( x + ǫ, x )) . Next, we prove that the three following statements hold.
Proposition 2.3
For any integers x , x and k the following holds for the strategies s k ( x , x ) .1. s k ( x , x ) is a sub-game perfect equilibrium in the game G k ( x , x ) .2. If a player does not compete in the first round of the game G k ( x , x ) , then it does not competein all subsequent rounds.3. The utility of the higher player in the game G k ( x , x ) is at least as large as the utility of thelower player. Essentially, we prove all three properties simultaneously by induction on the number of rounds ofthe game; to make the inductive argument easier to follow, we separate the three statements inSubsection A.1 of the appendix into three different claims. Let us mention two more claims thatwill be useful later on (all proofs are provided in Subsection A.2 of the appendix):
Claim 2.4 If u i ( s k ( x , x )) = kq , for some player i , then player i never competes in the game G k ( x , x ) . Claim 2.5
If player i competes in the first round of the game G k ( x , x ) and wins, then in thenext round of the game it also makes an offer to the stronger candidate. onnections to Urn Processes Note that since each player’s reputation is equal to the numberof stronger candidates it has hired, the reputations are always integers (assuming they start frominteger values). These integer values evolve while the players are competing; and once they stopcompeting, we know by statement (2) of Proposition 2.3 the exact outcome of the game since theplayers will behave exactly the same as in the game that this is its first round. This brings us tothe close resemblance between our recruiting game and a Polya Urn process [22].First, let us define what the Polya Urn process is:
Definition 2.6 (Polya Urn process)
Consider an urn containing b blue balls and r red balls.The process is defined over discrete rounds. In each round a ball is sampled uniformly at randomfrom the urn; hence the probability of drawing a blue ball is bb + r and the probability of drawing ared ball is rb + r . Then, the ball together with another ball of the same color are returned to the urn. There is a clear resemblance between our recruiting game and the urn model. As long as theplayers compete, their reputations evolve in the same way as the number of blue and red balls inthe urn, since the probabilistic rule for a candidate to select which firm to join is the same as therule for choosing which color to add to the urn, and by assumption the reputation of the winningplayer is increased by the stronger candidate’s quality, which is 1.A striking fact about urn models is that the fraction of the blue (or red) balls converges indistribution as the number of rounds goes to infinity. More specifically, if initially the urn containsa single red ball and a single blue ball then the fraction of blue balls converges to a uniformdistribution on [0 ,
1] as the number of rounds goes to infinity. More generally, the fraction of blueballs converges to the β distribution β ( b, r ). Understanding urn processes is useful for understandingour proofs; however we should stress that our model and its analysis have added complexity dueto the fact that players stop competing at a point in time that is strategically determined. Connections to Bandit Problems
It is interesting to note that as long as the lower player stayslower our equilibrium selection rule makes this effectively a one-player game. In a sense, the lowerplayer’s strategy in this phase resembles the optimal strategy in a mulit-armed bandit problem [11],and more specifically in a one-armed bandit problem [4]. In a one-armed bandit a single player isrepeatedly faced with two options (known as “arms” following the terminology of slot machines):the player can pull arm 1, which gives a reward sampled from some unknown distribution, orpull arm 2 which gives him a reward from a known distribution. Informally speaking, by pullingarm 1 the player gets both a reward and some information about the distribution from whichthe reward is drawn. The player’s goal is to maximize its expected reward possibly under somediscounting of future rounds. A celebrated result establishes that for some types of discounting(for example geometric) one can compute a number called the
Gittins index for each arm (basedon one’s observations and the prior) and the strategy maximizing the player’s expected reward isto pull the arm with the highest Gittins index in each round [11]. Since by definition the Gittinsindex of the fixed arm is fixed, this implies that once the Gittins index of the unknown arm dropsbelow the one of the known arm, the player should only pull the known arm. This also means thatthe player stops collecting information on the distribution of the unknown arm and hence from thisround onwards it always chooses the fixed arm.There are analogies as well as differences between our game and the one-armed bandit problem.In our game, the lower player is also faced with a choice between a risky option (competing)and a safe option (going for the weaker candidate). On the other hand, an important differencebetween our model and the one-armed bandit problem is that our game is in fact a two-player gameand at any point the lower-reputation player can become the higher-reputation one; this property8ontributes additional sources of complexity to the analysis of our game. Moreover, it is importantto note that for many distributions and discount sequences (including the ones most similar to ourgame) a closed-form expression of the Gittins index is unknown.
We begin by analyzing the game played over a fixed number of rounds k and study the dependenceof the performance ratio on k . In the next section, we turn to the main result of the paper, whichis to analyze the limit of the performance ratio as the number of rounds k goes to infinity.Our first result is a simple but powerful bound of q q on the performance ratio, which holdsfor all k . This is done by relating the performance ratio to players’ decision whether to compete inthe first round. The argument underlying this relationship is quite robust, in that it is essentiallybased only on the reasoning that the players can always decide to stop competing and go for theweaker candidate. This bound also implies that as q goes to 1 the performance ratio also goes to 1. Claim 3.1
The performance ratio of any game G k,q ( x , x ) is at least q q . Proof:
We begin with the simple observation that the expected social welfare equals the sumof the expected utilities of the two players in the beginning of the game. To get a lower boundon the performance ratio it is enough to compute an upper bound on the expected social welfare.This is done by observing that u i ( s k ( x , x )) ≥ kq , since a player can always secure a utility of kq by always making an offer to the weaker candidate. Hence, the following is a bound on theperformance ratio: u ( s k ( x , x )) + u ( s k ( x , x )) k (1 + q ) ≥ kqk (1 + q ) = 2 q q . Corollary 3.2
The performance ratio of any game G k,q ( x , x ) is at least / . The previous corollary holds for q > / q q > / q ≤ / / (1 + q ) ≥ / q < /
2, this is simply a single-round game. However, for q > / q − q + ǫ is an integer for an arbitrarily small ǫ >
0, a game of k q = q − q + ǫ roundsexhibits a performance ratio arbitrarily close to q q . It is interesting that the players’ strategies inthe games achieving this maximum performance ratio have a very specific structure – the playerscompete just for the first round and then the player who lost goes for the weaker candidate for therest of the game. Proposition 3.3
Let ǫ = ⌈ q − q ⌉ − q − q and k q = q − q + ǫ . Then, as ǫ approaches from above(remaining strictly positive), the performance ratio of the game G k q ,q ( x, x ) converges to q q . Proof:
Observe that by Claim 3.4 below the players in the game G k q ,q ( x, x ) compete for the firstround (since ǫ >
0) and then completely stop competing. Thus the expected social welfare of thecanonical equilibrium is k + ( k − q and its performance ratio is:1 + ( k − q ) k (1 + q ) = 1 + (( q − q + ǫ ) − q )( q − q + ǫ )(1 + q ) = 2 q + ǫ − ǫq q + q + ǫ − ǫq . It is not hard to see now that as ǫ approaches 0 the performance ratio approaches 2 q q .9e now prove for the k q ’s discussed in the previous proposition the players indeed competeonly for the first round and then stop competing. More formally we prove: Claim 3.4
In the game G k,q ( x, x ) for q − q < k ≤ − q the players compete in the first round andthen completely stop competing. Proof:
Player 2 (which is the lower player in the game) competes in the game G k,q ( x, x ) if:12 (1 + u ( s k − ( x, x + 1))) + 12 u ( s k − ( x + 1 , x )) > q + u ( s k − ( x + 1 , x )) . After some rearranging we get that this implies that player 2 competes if:1 + u ( s k − ( x, x + 1)) > q + u ( s k − ( x + 1 , x )) . Note that k ≤ − q = ⇒ q ≥ k − k . Thus, by Claim 3.5 below we have that for any x , x theplayers in the game G k − ,q ( x , x ) do not compete. This implies that u ( s k − ( x, x + 1)) = k − u ( s k − ( x + 1 , x )) = ( k − q . Thus, the players in the game G k,q ( x, x ) compete if k > ( k + 1) q implying q − q < k as required.Finally we prove: Claim 3.5 If q ≥ kk +1 then the players in the game G k,q ( x , x ) never compete. Proof:
Let x ≤ x . Player 2 competes in the game G k,q ( x , x ) if: x x + x (1 + u ( s k − ( x , x + 1))) + x x + x u ( s k − ( x + 1 , x )) > q + u ( s k − ( x + 1 , x )) . After some rearranging we get that player 2 competes if:1 + u ( s k − ( x , x + 1)) > x + x x q + u ( s k − ( x + 1 , x )) . Observe that u ( s k − ( x , x + 1)) ≤ k − k − u ( s k − ( x + 1 , x )) ≥ ( k − q and that by assumption x + x x ≥
2. Thus, we have that a necessary condition for player 2 to compete is that k > ( k + 1) q .This implies that for q ≥ kk +1 player 2 does not compete in the first round of the game G k,q ( x , x ).By part (2) of Proposition 2.3 we have that if a player does not compete in the first round of thegame it also does not compete in all subsequent rounds which completes the proof. A very similarproof works for the case that player 1 is the lower player. We now turn to the main question in the paper, which is the behavior of the performance ratio inthe limit as the number of rounds goes to infinity.Our main result here is that as k goes to infinity the performance ratio of the game G k (1 ,
1) goesto 1 + 2 qr q , where r = min { q, } . In particular for q < / k goes to infinity theperformance ratio goes to 1 + 2 q q . This function attains its minimum when q = √ . − ≈ . √ . ≈ . q ≥ /
2, on the other hand, this simply10mplies that as k goes to infinity the performance ratio of the game G k (1 ,
1) goes to 1. Defining r = min { q, } helps us to present a single unified proof both for q < / q ≥ / k the players compete goes to infinity. Henceforth, we will also refer to this probability as player’s2 relative reputation . We show that if the relative reputation of one of the players convergesto a number smaller than r , then after a fairly small number of rounds – specifically θ (ln( k )) –the players stop competing. The probability that the relative reputation of one of the playersconverges to something less than r is simply 2 r . Therefore, the expected social welfare of ourcanonical equilibrium converges to k + 2 qr ( k − θ (ln( k ))) and the performance ratio converges to qr q .We divide the proof to four subsections. In Subsection 4.1 we introduce t -binding games , whichgive us a formal way to study games in which the two players compete for at least the first t rounds.By showing that the utilities of the players in our game are at least as large as their utilities in the t -binding game we reduce our problem to showing that the expected utility in a t -binding game is“large enough”. This is done in Subsection 4.3. The proof relies on Subsection 4.2 which, looselyspeaking, shows that if after t rounds of competition the relative reputation of the lower player isnon-trivially smaller than r then the lower player stops competing. Finally, in Subsection 4.4 westate the formal theorem and wrap up the proof. t -Binding Games A recruiting game is t -binding if in the first t rounds the two players are required to compete forthe stronger candidate. We denote a t -binding game by G tk ( x , x ). We also denote by s tk ( x , x )the canonical equilibrium of the game G tk ( x , x ) in which the players compete for the first t roundsand then follow the strategies s k − t ( x ′ , x ′ ) in the resulting game.Denote by u ( s tk ( x , x )) the expected social welfare of the canonical equilibrium in the game G tk ( x , x ). It is intuitive to suspect that making the players compete for the first t rounds canonly decrease their utility. In the next lemma we prove that this intuition is indeed correct: Lemma 4.1
The expected social welfare of the game G k (1 , is greater than or equal to the expectedsocial welfare of the game G tk (1 , ; that is, u ( s k (1 , ≥ u ( s tk (1 , . Proof:
We prove the lemma by proving a stronger claim:
Claim 4.2
The expected utility of each of the players in the game G tk (1 , for ≤ t < k is greaterthan or equal to their expected utility in the game G t +1 k (1 , . Proof:
For simplicity we prove the claim for player 2; however the claim holds for both players.By definition, in the game G tk (1 ,
1) the players compete for at least the first t rounds. Duringthis phase of competition, the two players’ reputations evolve according to the update rule for astandard Polya urn process, as described in Section 2. A standard result on that process impliesthat at the end of these t rounds with probability t +1 player 1 has a reputation of 1 + t − i andplayer 2 has a reputation of 1 + i for 0 ≤ i ≤ t . Thus, we have that: u ( s tk (1 , t + 1 t X i =0 u ( s k − t (1 + t − i, i ))11et I δ = { i | f ( s k − t (1 + t − i, i )) = δ } for δ ∈ {h + , + i , h + , −i , h− , + i} . For example, I h + , + i is theset of all indices i for which the players compete in the first round of the game G k − t (1 + t − i, i ).We can now write the sum, usefully, as u ( s tk (1 , t + 1 X i ∈ I h + , + i u ( s k − t (1 + t − i, i )) + 1 t + 1 X i ∈ I h + , −i u ( s k − t (1 + t − i, i ))+ 1 t + 1 X i ∈ I h− , + i u ( s k − t (1 + t − i, i ))By this partition: • For i ∈ I h + , + i , we have u ( s k − t (1 + t − i, i )) = u ( s k − t (1 + t − i, i )) – since in both ofthese games the two players compete in the first round. • For i ∈ I h + , −i , we have u ( s k − t (1 + t − i, i )) ≥ u ( s k − t (1 + t − i, i )) – since u ( s h + , −i k − t (1 + t − i, i )) ≥ u ( s h + , + i k − t (1 + t − i, i )). (in the first round of the game player 2 prefers goingafter the weaker candidate over competing). • For i ∈ I h− , + i , we have u ( s k − t (1 + t − i, i )) > u ( s k − t (1 + t − i, i )) – since u ( s h− , + i k − t (1 + t − i, i )) = 1+ u ( s k − t − (1+ t − i + q, i + 1)) > u ( s h + , + i k − t (1+ t − i, i )) by monotonicity.Thus, we have u ( s tk (1 , ≥ t +1 P ti =0 u ( s k − t (1 + t − i, i )) = u ( s t +1 k (1 , This next phase of our analysis is composed of two parts: in the first part we show that the utilityof the lower player in a k -round game is upper bounded by max { b q ( k, t, x ) , kq } for some function b q ( · ) to be later defined. In the second part we compute the conditions under which b q ( k, t, x ) < kq which implies that under the same conditions the lower player in the game stops competing.For this subsection we denote player 1’s reputation after t rounds by t − x and player 2’sreputation by x . Both statements below also hold for player 1 and the game G k ( x, t − x ).The following notation will be useful for our proofs: • f q ( i, t ) = (cid:0) ti (cid:1) q i (1 − q ) t − i – probability mass function for the binomial distribution with t trials. • F q ( x, t ) = P xi =0 (cid:0) ti (cid:1) q i (1 − q ) t − i – cumulative distribution function for an integer x .The function that we use to upper bound the player’s utility is: b q ( k, t, x ) = xt + 3 F r ( x, t ) k + (1 − F r ( x, t ))( k − q = xt + ( k − q + 3 F r ( x, t ) · (cid:0) ( k − − q ) + 1 (cid:1) To understand the intuition behind the upper bound function b q ( k, t, x ) it is useful to look atan alternative description of the urn process. Under this description, we have a coin whose biasis sampled from a uniform distribution on [0 , r (recall that r = min { q, } ). We refer to the event inwhich the bias of the coin is greater than r as a good event, and the event it is not a bad event.To upper-bound the player’s utility we assume that if the good event happens the player wins thestronger candidate for all subsequent rounds and hence its utility is k . If the bad event happensthen the player completely stops competing and thus its utility is ( k − q .We show that max { b q ( k, t, x ) , kq } is indeed an upper bound on the players’ utility as the previousintuition suggests. Lemma 4.3
For any k , x and t > / − r ) , we have u ( s k ( t − x, x )) ≤ max { b q ( k, t, x ) , kq } . Proof:
We divide the proof into two cases. When, r ≤ x + 1 t + 1 the bound we need to prove is veryloose and hence we can prove it directly. However, for r > x + 1 t + 1 proving this bound is more trickyand for this we use an induction that some times relies on the first case. The proofs of these twocases are provided in Claim B.1 and Claim B.2 of the appendix.We can now use the previous bound to compute the conditions under which the lower playerprefers to stop competing. Theorem 4.4
In the game G k ( t − p · t, p · t ) for p = r − ǫ , ǫ > and t = max { / − r ) , k ) − ln( q − p )( r − p ) } player does not compete at all. Proof:
By Lemma 4.3 we have that u ( s k ( t − p · t, p · t )) ≤ max (cid:8) b q ( k, t, p · t ) , kq (cid:9) for t > / − r ) .Since we have that u ( s k ( t − p · t, p · t )) ≥ kq , if we show that b q ( k, t, p · t ) ≤ kq , then we will have u ( s k ( t − p · t, p · t )) = kq . It will then follow from Claim 2.4 that the lower player (player 2) does notcompete at all. The theorem will thus follow if we show that for t = max { / − r ) , k ) − ln( q − p )( r − p ) } ,we have b q ( k, t, p · t ) ≤ kq .By Hoeffding’s inequality with ǫ = r − p , we get that F r ( p · t, t ) ≤ e − t ( r − p ) . Now, to computethe value of t for which u ( s k ( t − p · t, p · t )) = kq , we simply find the value of t for which thefollowing inequality holds: p + ( k − q + 3 e − t ( r − p ) (cid:0) ( k − − q ) + 1 (cid:1) ≤ kq After some rearranging we get that:3 e − t ( r − p ) (cid:0) ( k − − q ) + 1 (cid:1) ≤ q − p k − − q ) + 1 ≤ e t ( r − p ) ( q − p )Taking natural logarithms we get:ln(3( k − − q ) + 1) ≤ t ( r − p ) + ln( q − p )ln(3( k − − q ) + 1) − ln( q − p )2( r − p ) ≤ t In particular this implies that the claim holds for t ≥ k ) − ln( q − p )( r − p ) .13 .3 The Expected Social Welfare of a t -Binding Game We show that for large enough k the social welfare of the t -binding game G tk (1 ,
1) is relatively high.This is done by showing that there exists some t , such that after competing for t rounds, withprobability 2( r − ǫ ) − t +1 the players reach a game in which the lower player (either player 1 orplayer 2) does not want to compete any more. Lemma 4.5
For every ǫ > and k ≥ e r − ǫ ) ǫ + e / − r ) , there exists t such that the expected socialwelfare of the t -binding game G tk (1 , is at least k · (cid:0) q ( r − ǫ − ǫ ) (cid:1) . Proof:
By the assumption that the game is t -binding we have that both players compete overthe stronger candidate for the first t rounds. This implies that at the end of these t rounds withprobability t +1 player 1 has a reputation of 1 + t − i and player 2 has a reputation of 1 + i for0 ≤ i ≤ t . Or, in other words, the relative reputation of player 2 is it +2 with probability t +1 .Notice that for any 0 ≤ i ≤ ⌊ ( r − ǫ )( t + 2) ⌋ − it +2 < r − ǫ . Thus, the probabilitythat the relative reputation of player 2 is smaller than ( r − ǫ ) is1 t + 1 · ( ⌊ ( r − ǫ )( t + 2) ⌋ − ≥ t + 1 · (( r − ǫ )( t + 2) − ≥ ( r − ǫ ) − t + 1This implies that with probability of at least ( r − ǫ ) − t +1 after t rounds the current game is G k − t (( t + 2)(1 − p ) , p · ( t + 2)) for p < r − ǫ . Notice that by symmetry the same holds for player1. By choosing t that obeys the requirements of Theorem 4.4 we get that the lower player in thisgame does not compete. Therefore, the probability that one of the players stops competing after t rounds is at least 2( r − ǫ ) − t +1 . To bound the expected social welfare we make the conservativeassumption that with probability 1 − (2( r − ǫ ) − t +1 ) the players compete till the end of the gameand get that: u ( s tk (1 , ≥ k + 2 q (cid:18) ( r − ǫ ) − t + 1 (cid:19) ( k − t ) ≥ k + 2 q ( r − ǫ ) k − q ( r − ǫ ) t − kqt Next, we show that for k ≥ e r − ǫ ) ǫ + e / − r ) and t = k ) − ln( ǫ ) ǫ the conditions for both Theorem4.4 and this Lemma hold. Indeed, if Theorem 4.4 holds, we have that for every 0 < p < r − ǫ theplayers in the game G k − t (( t + 2)(1 − p ) , p · ( t + 2)) for p < r − ǫ do not compete, as required.Recall that Theorem 4.4 requires t + 2 to be at least max { / − r ) , k ) − ln( q − p )( r − p ) } . Observe that k ) − ln( ǫ ) ǫ ≥ k ) − ln( q − p )( r − p ) as by definition q − p ≥ r − p > ǫ ; and that since ln( k ) > / − r ) wealso have that t ≥ / − r ) .Next, we show that u ( s tk (1 , ≥ k · (cid:0) q ( r − ǫ − ǫ ) (cid:1) . We begin by plugging in t = k ) − ln( ǫ ) ǫ : u ( s tk (1 , ≥ k + 2 kq ( r − ǫ ) − q ( r − ǫ ) · k ) − ln( ǫ ) ǫ − kq k ) − ln( ǫ ) ǫ > k + 2 kq ( r − ǫ ) − q ( r − ǫ ) · k ) − ln( ǫ ) ǫ − kqǫ ln( k ) ≥ k + 2 kq ( r − ǫ − ǫ ) − q ( r − ǫ ) · k ) ǫ + 2 q ( r − ǫ ) ln( ǫ ) ǫ To prove the Lemma we show that for k ≥ e r − ǫ ) ǫ + e / − r ) the following two inequalities hold:14. ( r − ǫ ) · k ) kǫ < ǫ : For this we do a variable substitution and denote ln( k ) = z , so that k = e z .Now we find z such that 4( r − ǫ ) z < ǫ · e z . By Taylor expansion we have that e z > z .Thus, we can instead compute when 4( r − ǫ ) z < ǫ · z and get that the inequality holds for z > r − ǫ ) ǫ . This implies that the inequality holds for k > e r − ǫ ) ǫ .2. | ( r − ǫ ) ln( ǫ ) kǫ | < ǫ : This condition also holds for k > e r − ǫ ) ǫ since if k > e r − ǫ ) ǫ by Taylorexpansion we have that k > (( r − ǫ ) ǫ ) ) / r − ǫ ) ǫ > r − ǫǫ ·| ln( ǫ ) | and therefore | ( r − ǫ ) ln( ǫ ) kǫ | <ǫ .Thus, for k ≥ e r − ǫ ) ǫ + e / − r ) and t = k ) − ln( ǫ ) ǫ we have that u ( s tk (1 , ≥ k · (cid:0) q ( r − ǫ − ǫ ) (cid:1) as required. Theorem 4.6
For ǫ > and k ≥ e r − ǫ ) ǫ + e / − r ) , the performance ratio of the game G k (1 , isat least q ( r − ǫ − ǫ )1+ q . Proof:
By Lemma 4.1 we have that for any t , u ( s k (1 , ≥ u ( s tk (1 , t such that u ( s tk (1 , ≥ k (cid:0) q ( r − ǫ − ǫ ) (cid:1) . By combining the two weget that u ( s k (1 , ≥ k (cid:0) q ( r − ǫ − ǫ ) (cid:1) . This means that the performance ratio of the game G k (1 ,
1) is at least k ( q ( r − ǫ − ǫ ) ) k (1+ q ) = q ( r − ǫ − ǫ )1+ q . Corollary 4.7 As k goes to infinity, the performance ratio of the game G k (1 , , ) goes to rq q . When firms compete for job applicants over many hiring cycles, there is a basic strategic tensioninherent in the process: trying to recruit highly sought-after job candidates can build up a firm’sreputation, but it comes with the risk that firm will fail to hire anyone at all. In this paper, we haveshown how this tension can arise in a simple dynamic model of job-market matching. Althoughour model is highly stylized, it contains a number of interesting effects that we analyze, includingthe way in which competition can lead to inefficiency through underemployment (quantified in ouranalysis of the performance ratio at equilibrium) and the possibility of different modes of behavior,in which a weaker firm may end up competing forever, or it may give up at some point and acceptits second-place status.The model and analysis also suggest a number of directions for further investigation. Onedirection is to vary the competition function that determines the outcome of a competition betweenthe two firms when they make offers to the same candidate. As noted above, this can be viewed asvarying the way in which candidates make decisions between firms based on their reputations. InSection C of the appendix, we explore this issue by considering an alternate rule for competition inwhich the lower-reputation player wins with a fixed probability p < (independent of the differencein reputation) and the higher-reputation player wins with probability 1 − p .This fixed-probability competition function is simpler in structure than the Tullock function,and it is illuminating in that it cleanly separates two different aspects of the strategic decisionbeing made about future rounds. With the Tullock function, when the lower player competes, ithas the potential for a short-term gain in its success probability even in the next round (since the15atio of reputations will change), and it also has the potential for a long-term gain by becoming thehigher player. With the fixed-probability competition function, the short-term aspect is effectivelyeliminated, since as long as a player remains the lower party, it has the same probability of success;we are thus able to study strategic behavior about competing when the only upside is the long-rangeprospect of becoming the higher player. We show that the performance is generally much betterwith this fixed-probability rule than with the Tullock function, providing us with further insightinto the specific way in which competition leads to inefficiency through a reduced performanceratio.Other directions that lead quickly to interesting questions are to consider the case of morethan two firms, and to consider models in which the candidates have different characteristics indifferent time periods. For both of these general directions, our initial investigations suggest thatthe techniques developed here will be useful for shedding light on the properties of more complexmodels that take these issues into account. Acknowledgments
We thank Itai Ashlagi, Larry Blume, Shahar Dobzinski, Bobby Kleinberg,and Lionel Levine for very useful suggestions and references.
References [1] R. Albert and A.-L. Barab´asi. Statistical mechanics of complex networks.
Rev. Modern Physics ,74(2002).[2] A. Beggs and P. Klemperer. Multi-period competition with switching costs.
Econometrica
Review of EconomicStudies , 54(1987).[4] D. Berry, B. Fristedt. Bernoulli one-armed bandits: arbitrary discount sequences.
Ann. Stat. ,7(1979).[5] A. Broder, A. Kirsch, R. Kumar, M. Mitzenmacher, E. Upfal, S. Vassilvitskii. The hiringproblem and lake wobegon strategies.
SIAM J. Comp. , 39(2009).[6] J. Chen, U. Doraszelski, J. Harrington. Avoiding market dominance: product compatibilityin markets with network effects.
RAND J. Econ. , 40(2009).[7] J. S. Coleman. Matching processes in the labor market.
Acta Sociologica
Proc. 26th AAAI Conf. , 2012.[9] E. Drinea, A. Frieze, M. Mitzenmacher. Balls and bins models with feedback.
Proc. ACM-SIAM SODA , 2002.[10] W. Feller.
An Introduction to Probability Theory and Its Applications , volume 1. Wiley, 1968.[11] J. Gittins, D. Jones. A dynamic allocation index for the sequential design of experiments. InJ. Gani, editor,
Progress in Statistics , North-Holland, 1974.[12] M. Granovetter.
Getting a Job: A Study of Contacts and Careers . University of Chicago Press,1974. 1613] G. Haeringer and M. Wooders. Decentralized job matching.
Intl. J. Game Theory , 40(2011).[14] K. Hazma. The smallest uniform upper bound on the distance between the mean and medianof the binomial and Poisson distributions.
Stat. Prob. Let. , 23(1995).[15] N. Immorlica, A. Kalai, B. Lucier, A. Moitra, A. Postlewaite, M. Tennenholtz. Dueling algo-rithms.
ACM STOC , 2011.[16] N. Immorlica, R. Kleinberg, M. Mahdian. Secretary problems with competing employers.
WINE , 2006.[17] B. Jun and X. Vives. Strategic incentives in dynamic duopoly.
Journal of Economic Theory .116(2004).[18] M. Mitzenmacher. A brief history of generative models for power law and lognormal distribu-tions.
Internet Math. , 2004.[19] D. Mortensen and C. Pissarides. New developments in models of search in the labor market.In
Handbook of Labor Economics , 1999.[20] S. Mullainathan, J. Schwartzstein, A. Shleifer. Coarse thinking and persuasion.
Q. J. Econ. ,123(2008).[21] A. J. Padilla. Revisiting Dynamic Duopoly with Consumer Switching Costs.
Journal of Eco-nomic Theory . 67(1995), Pages 520530.[22] R. Pemantle. A survey of random processes with reinforcement.
Probability Surveys , 4:1–79,2007.[23] Lauren A. Rivera. Hiring as Cultural Matching: The Case of Elite Professional Service Firms.
American Sociological Review
J. Econ.Lit. , 43(2005).[25] A. Roth, M. Sotomayor.
Two-sided matching: A study in game-theoretic modeling and analysis .Cambridge University Press, 1990.[26] R. Shimer and L. Smith. Assortative matching and search.
Econometrica , 68(2):343–369, Mar.2000.[27] S. Skaperdas. Contest success functions.
Econ. Th. , 7(1996).[28] G. Tullock. Efficient rent seeking. In
Towards a theory of the rent-seeking society , 1980.[29] D. Turban and D. Cable. Firm reputation and applicant pool characteristics.
J. Organiz.Behav.
24, 733751 (2003).[30] J. Zou, S. Gujar, and D. Parkes. Tolerable manipulability in dynamic assignment withoutmoney.
Proc. 24th AAAI , 2010. 17
The canonical equilibrium and its Properties (includes proofsfrom Section 2)
Our main goal in this section is to prove that the strategies s k ( x , x ) form a subgame perfectequilibrium in the game G k ( x , x ), and to present the proofs of some useful properties of thisequilibrium. The arguments can be carried out in a setting more general than that of the Tullockcompetition function, and we present them for a broader class of competition functions, specifyingthe probability that a candidate chooses each firm in the event of competition between them. Wework at this greater level of generality for two reasons. First, it makes clear what properties of thecompetition function are necessary for the equilibrium results. Second, and more concretely, westudy a variant of the model in Section C that involves a different competition function, in whicha candidate picks the lower-reputation firm with a fixed probability p < regardless of the actualnumerical values of the reputations. This fixed-probability competition function satisfies our moregeneral assumptions, and thus we can apply all the results of this section to it.For ease of exposition, the results in Section 2 of the main text are presented specifically for theTullock competition function; as a result, to complete the link back to this section, we state whichclaims here generalize each claim from Section 2.Our results hold for the following general definition of a competition function c : R × R → (0 , c ( x , x ) should represent the probability that player 1 wins acompetition when the two players’ strengths are x and x respectively. Definition A.1
A function c : R × R → (0 , is a competition function if: • c ( x , x + ǫ ) ≤ c ( x , x ) ≤ c ( x + ǫ, x ) . • For every x = x : c ( x , x ) = (1 − c ( x , x )) . • c ( x, x ) ≥ (1 − c ( x, x )) . With this notation in mind the utility of player 2 for competing is now:(1 − c ( x , x )) · (1 + u ( s k − ( x , x + 1))) + c ( x , x ) · u ( s k − ( x + 1 , x )) . Observe that the following two properties hold for any competition function. These will beuseful for later proofs: • Let ǫ >
0. (1 − c ( x , x + ǫ )) ≥ c ( x , x ). • Let ǫ >
0. (1 − c ( x , x )) ≥ c ( x , x + ǫ ).To see why the first statement holds, observe that if x = x + ǫ we have that:(1 − c ( x , x + ǫ )) = c ( x + ǫ, x ) ≥ c ( x , x ) . Else, we have that x = x and in this case we have that:(1 − c ( x , x + ǫ )) ≥ (1 − c ( x , x )) = c ( x , x ) . The second statement also holds for similar reasons.We begin by showing that since the lower player in s k ( x , x ) chooses the strategy maximizingits utility it can always guarantee itself a utility of at least kq by always going for the weakercandidate. 18 laim A.2 Let player i be the lower player in the game G k ( x , x ) . Then u i ( s k ( x , x )) ≥ kq : Proof:
We prove the claim for the case that x ≤ x but a similar proof can be easily devisedfor the case that x < x . We prove the claim by induction on the number of rounds k . For thebase case k = 1, it is easy to see that the utility of player 2 is max { (1 − c ( x , x )) , q } , and thereforethe claim holds. We assume correctness for ( k − k -round games.Observe that: u ( s k ( x , x )) = max { u ( s h + , + i k ( x , x )) , q + u ( s k − ( x + 1 , x )) } . Since player 2 isalso the lower player in the game G k − ( x + 1 , x ) we can use the induction hypothesis and getthat u ( s k − ( x + 1 , x )) ≥ ( k − q which completes the proof.Next we show that the utilities u i ( s k ( x , x )) are monotone increasing in player i ’s reputationand monotone decreasing in its opponent’s reputation. Claim A.3 (generalizes Claim 2.2)
For any x , x , and ǫ > :1. u ( s k ( x + ǫ, x )) ≥ u ( s k ( x , x )) and u ( s k ( x + ǫ, x )) ≤ u ( s k ( x , x )) u ( s k ( x , x − ǫ )) ≥ u ( s k ( x , x )) and u ( s k ( x , x − ǫ )) ≤ u ( s k ( x , x )) . Proof:
We prove both properties concurrently by induction over k , the number of rounds in thegame. Since the proofs for both properties are very similar, we only present here the proof for thefirst property. For the base case, k = 1, we distinguish between the following cases cases:1. s ( x , x ) = s ( x + ǫ, x ) : if they compete in in both s ( x , x ) and s ( x + ǫ, x ), then theclaim holds simply because c ( x + ǫ, x ) ≥ c ( x , x ) and (1 − c ( x + ǫ, x )) ≤ (1 − c ( x , x )).Else, in both games the players have the exact same utility (either 1 or q ).2. s ( x , x ) = s ( x + ǫ, x ), s ( x , x ) = h + , + i and s ( x + ǫ, x ) = h + , + i : observe that thisis only possible if x < x ≤ x + ǫ and in this case we have that s ( x , x ) = h− , + i and s ( x + ǫ, x ) = h + , −i so it is not hard to see that the claim holds.3. s ( x , x ) = h + , + i and s ( x + ǫ, x ) = h + , + i : this implies that x ≥ x since the utilityplayer 1 can get for competing in G ( x + ǫ, x ) is greater than the utility it can get forcompeting in G ( x , x ): c ( x + ǫ, x ) ≥ c ( x , x ) > q . Therefore, u ( s ( x + ǫ, x )) ≥ u ( s ( x , x )). For player 2, u ( s k ( x + ǫ, x )) ≤ u ( s k ( x , x )), since we have that q < (1 − c ( x , x )).4. s ( x , x ) = h + , + i and s ( x + ǫ, x ) = h + , + i : this implies that x > x . As for player 2 itsnot hard to see that: u ( s h + , + i ( x , x )) ≥ u ( s h + , + i ( x + ǫ, x )). Thus the only reason thatthe players do not compete in s ( x , x ) is that player 1 prefers to go for the weaker candidateand it is entitled to make this choice in s ( x , x ) only is x < x . It is not hard to see thatthe claim holds for this case as well.We now assume that both statements 1 and 2 hold for ( k − k -round games. The proof takes a very similar structure to the proof for the basecase, except now we shall use the induction hypothesis instead of first principles. The followingobservation will be useful for the proof: Observation A.4
By applying the induction hypothesis we get that the following two statementshold for any δ ∈ {h + , + i , h + , −i , h− , + i} :1. u ( s δk ( x + ǫ, x )) ≥ u ( s δk ( x , x )) 19 . u ( s δk ( x , x )) ≥ u ( s δk ( x + ǫ, x ))Take for example the first statement and consider δ = h + , + i . To see why it is indeed the casethat u ( s h + , + i k ( x + ǫ, x )) ≥ u ( s h + , + i k ( x , x )), observe that by the induction hypothesis we havethat: u ( s k − ( x + 1 + ǫ, x )) ≥ u ( s k − ( x + 1 , x )), u ( s k − ( x + ǫ, x + 1)) ≥ u ( s k − ( x , x + 1)), u ( s k − ( x + ǫ + 1 , x )) ≥ u ( s k − ( x + ǫ, x + 1)) and that c ( x + ǫ, x ) ≥ c ( x , x ). Similarly,we can use the induction hypothesis to prove that the two statments are correct for any δ ∈{h + , + i , h + , −i , h− , + i} .Just as in the base case, we now distinguish between the following cases:1. f ( s k ( x , x ))) = f ( s k ( x + ǫ, x )) : the claim holds by Observation A.4 above.2. f ( s k ( x , x )) = f ( s k ( x + ǫ, x )), f ( s k ( x , x )) = h + , + i and f ( s k ( x + ǫ, x )) = h + , + i :observe that this is only possible if x < x ≤ x + ǫ and in this case we have that f ( s k ( x , x )) = h− , + i and f ( s k ( x + ǫ, x )) = h + , −i so it is not hard to see that the claimholds.3. f ( s k ( x , x )) = h + , + i and f ( s k ( x + ǫ, x )) = h + , + i : Similar to the corresponding casefor k = 1, observe that this implies that x ≥ x . As by Observation A.4 we have that u ( s h + , + i k ( x + ǫ, x )) ≥ u ( s h + , + i k ( x , x )). Thus, if the players do not compete in s k ( x + ǫ, x ) it can only be because the lower player prefers not to compete and this lower playerhas to be player 2. Now, by applying the induction hypothesis for player 1 we get that u ( s k − ( x + ǫ + 1 , x )) ≥ u ( s k − ( x + 1 , x )) ≥ u ( s k − ( x , x + 1)). Thus, it is not hard tosee that u ( s k ( x + ǫ, x )) ≥ u ( s k ( x , x )). For player 2, since as the lower player it choosesto compete in f ( s k ( x , x )) but not in f ( s k ( x + ǫ, x )) we have that: u ( s k ( x , x ) = u ( s h + , + i k ( x , x )) > u ( s h + , −i k ( x , x )) ≥ u ( s h + , −i k ( x + ǫ, x )) = u ( s k ( x + ǫ, x )where the last transition is by Observation A.4.4. f ( s k ( x , x )) = h + , + i and f ( s k ( x + ǫ, x )) = h + , + i : this is similar to the previous caseonly now we have that x < x . The reason is that by Observation A.4 we have that u ( s h + , + i k ( x , x )) ≥ u ( s h + , + i k ( x + ǫ, x )). Therefore the lower player in the game G k ( x , x )is player 1. After establishing this, it is easy to verify that the claim holds by applying theinduction hypothesis in a very similar manner to the previous case. A.1 s k ( x , x ) is a Subgame Perfect Equilibrium in the Game G k ( x , x ) We prove the following three statements simultaneously by induction on the number of rounds inthe game:
Proposition A.5 (generalizes Claim 2.3)
For any integers x , x and k the following holds forthe strategies s k ( x , x ) .1. s k ( x , x ) is a sub-game perfect equilibrium in the game G k ( x , x ) .2. If a player does not compete in the first round of the game G k ( x , x ) , then it does not competein all subsequent rounds. . The utility of the higher player in the game G k ( x , x ) is at least as large as the utility of thelower player. We separate the simultaneous induction into three numbered statements above. Proving thesethree statements by simultaneous induction on k means that in studying properties of s k ( x , x )and G k ( x , x ), we can assume that all three parts hold for s k ′ ( x , x ) and G k ′ ( x , x ) for every0 < k ′ ≤ k − s k ( x , x ) is asub-game perfect equilibrium. As part of the induction the proof relies on the correctness of parts2 and 3 for games of less than k rounds. Next, we prove parts 2 and 3 in Claim A.7 and PropositionA.9 respectively. Both proofs assume that s k ′ ( x , x ) is a subgame perfect equilibrium for every0 < k ′ ≤ k − Proposition A.6
The strategies described by s k ( x , x ) are a subgame perfect equilibrium in thegame G k ( x , x ) for every two integers x , x . Proof:
We prove the claim by induction on k the number of rounds. We only present the prooffor x ≥ x as the proof for the case that x < x is very similar. For the base case k = 1, if c ( x , x ) > q , then it is clearly the case that player 1 competes. Else, since x ≥ x we have that(1 − c ( x , x )) ≤ c ( x , x ) ≤ q , therefore player 2 does not want to compete as well. As clearlyplayer 1 cannot benefit from competing over the weaker candidate, this implies that there existsan equilibrium in which player 1 goes for the stronger candidate.Next, we assume correctness for k ′ -round games for any 0 < k ′ ≤ k − k -roundgames. This means we can apply Corollary A.8 and get that once the lower player prefers to gofor the weaker candidate it completely stops competing and Proposition A.9 to get that the higherplayer always has greater utility than the lower player.Assume towards contradiction that in the unique equilibrium for the first round of the game G k ( x , x ) player 1 goes after the weaker candidate and player 2 goes after the stronger candidate.Thus, player 1’s utility is u ( s h− , + i k ( x , x )) = q + u ( s k − ( x , x + 1)).We first observe that by monotonicity if player 1 prefers to go for the weaker candidate overcompeting, it has to be the case that c ( x , x ) < q . We also observe that it has to be the casethat f ( s k ( x , x )) = h + , + i . As it is easy to see that in the case of f ( s k ( x , x )) = h + , −i player 1prefers to go for the stronger candidate over competing for the weaker candidate. We now distinguish between 3 possible scenarios in s k − ( x , x + 1) which is by the inductionhypothesis a subgame perfect equilibrium in G k − ( x , x + 1):1. f ( s k − ( x , x + 1)) = h + , + i : observe that in this case player 1 prefers to compete in the firstround of the game G k ( x , x ) since by monotonicity u ( s k − ( x +1 , x +1)) ≥ u ( s k − ( x , x +2)) ; by the induction hypothesis we have that s k − ( x , x + 1) and s k − ( x + 1 , x ) aresubgame perfect equilibria we have that u ( s k − ( x + 1 , x )) ≥ q + u ( s k − ( x + 1 , x + 1))and u ( s k − ( x , x + 1)) ≥ q + u ( s k − ( x , x + 2)); and c ( x , x ) ≥ c ( x , x + 1). Thus, wehave that: u ( s h + , + i k ( x , x )) = c ( x , x ) · (1 + u ( s k − ( x + 1 , x ))) + (1 − c ( x , x )) · u ( s k − ( x , x + 1)) ≥ c ( x , x ) · (1 + ( q + u ( s k − ( x + 1 , x + 1)))) + (1 − c ( x , x )) · ( q + u ( s k − ( x , x + 2))) ≥ q + c ( x , x + 1) · (1 + u ( s k − ( x + 1 , x + 1))) + (1 − c ( x , x + 1)) · u ( s k − ( x , x + 2))= q + u ( s h + , + i k − ( x , x + 1)) = u ( s h− , + i k ( x , x )) Observe that this holds since q + u ( s k − ( x + q, x )) < u ( s k − ( x + 1 , x )) by monotonicity. This is a welldefined use of Claim A.3 since it does not assume the players’ reputations to be integers. To handle the case of k = 2 we define u i ( s ( x , x )) = 0. h− , + i is not an equilibrium for the first round of the game G k ( x , x ).2. f ( s k − ( x , x + 1)) = h + , −i : this implies that x ≥ x + 1. We can apply Corollary A.8 andget that for player 2, u ( s k − ( x , x + 1)) = ( k − q . Since by monotonicity we have that( k − q = u ( s k − ( x , x + 1)) ≥ u ( s k − ( x + 1 , x )) we get that: u ( s h + , + i k ( x , x )) ≤ (1 − c ( x , x )) + ( k − q < kq Where the last transition is due to the fact that (1 − c ( x , x )) ≤ c ( x , x ) < q . Hence,by Claim A.2 the lower player (player 2) does not want to compete in the game G k ( x , x ).Therefore h− , + i is not the unique equilibrium for the first round of the game G k ( x , x ).3. f ( s k − ( x , x + 1)) = h− , + i : This implies that x = x therefore we can apply Corollary A.8for player 1 in the game G k − ( x , x +1) and get that u ( s h− , + i k ( x , x )) = q + u ( s k − ( x , x +1)) = kq . We now have the following chain of inequalities, by applying Proposition A.9: u ( s h− , + i k ( x , x )) > u ( s h + , + i k ( x , x )) = u ( s k ( x , x )) ≥ u ( s k ( x , x )) ≥ kq The last transition is due to the fact that player 2 is the lower player in the game G k ( x , x )and thus we can use Claim A.2. This is of course in contradiction to the fact that u ( s h− , + i k ( x , x )) = kq . A.1.1 Proof of Part (2) of Proposition A.5
We show that if a player prefers not compete in the first round of the game G k ( x , x ), then it doesnot compete in all subsequent rounds. This is done by showing that if a player does not competein the first round of a game, then it does not compete in the second round. The proof assumesthat s k − ( x , x ) is a subgame perfect equilibrium as it used as part of the induction.Formally we show: Claim A.7 If s k − ( x , x ) is a subgame perfect equilibrium for every x and x , then • f ( s k ( x , x )) = h + , −i = ⇒ f ( s k − ( x + 1 , x )) = h + , −i . • f ( s k ( x , x )) = h− , + i = ⇒ f ( s k − ( x , x + 1)) = h− , + i . Proof:
We prove the first statement of the claim as the proof of the second statement is verysimilar. Assume towards a contradiction that f ( s k ( x , x )) = h + , −i but f ( s k − ( x + 1 , x )) = h + , −i . The fact that f ( s k ( x , x )) = h + , −i implies that x ≤ x ; thus if f ( s k − ( x + 1 , x )) = h + , −i it has to be the case that f ( s k − ( x + 1 , x )) = h + , + i . Therefore, player 2’s utility is: u ( s k ( x , x )) = q + (1 − c ( x + 1 , x )) · (1 + u ( s k − ( x + 1 , x + 1))) + c ( x + 1 , x ) · u ( s k − ( x + 2 , x ))Observe that the following holds:1. f ( s k − ( x + 1 , x )) = h + , + i = ⇒ u ( s k − ( x + 1 , x )) > q + u ( s k − ( x + 2 , x )).2. u ( s k − ( x , x + 1)) ≥ q + u ( s k − ( x + 1 , x + 1)).22he first of these statements holds since player 2 is the lower player in the game G k − ( x +1 , x ), andthe second statement holds since by assumption s k − ( x , x + 1) is a subgame perfect equilibrium.Thus, we have that: u ( s h + , + i k ( x , x )) > q + (1 − c ( x , x )) · (1 + u ( s k − ( x + 1 , x + 1))) + c ( x , x ) · u ( s k − ( x + 2 , x ))This implies that u ( s h + , + i k ( x , x )) > u ( s k ( x , x )) in contradiction to the assumption that player2 maximizes its utility by first going for the weaker candidate ( f ( s k ( x , x )) = h + , −i ). Corollary A.8
For x ≤ x , if s k ′ ( x , x ) is a subgame perfect equilibrium for every < k ′ ≤ k − , x and x , then, f ( s k ( x , x )) = h + , −i = ⇒ u ( s k ( x , x )) = kq . A.1.2 Proof of Part (3) of Proposition A.5
We show that the utility of the higher player in the game G k ( x , x ) is at least as large as theutility of the lower player. This is based on Claim A.10 and Claim A.11 below. Proposition A.9 If s k − ( x , x ) is a subgame perfect equilibrium for every two integers x and x then: • For x ≥ x : u ( s k ( x , x )) ≥ u ( s k ( x , x )) . • For x > x : u ( s k ( x , x )) ≥ u ( s k ( x , x )) . Proof: • For x ≥ x . By Claim A.10 we have that u ( s k ( x , x )) ≥ u ( s k ( x , x )). Now, by mono-tonicity we have that u ( s k ( x , x )) ≥ u ( s k ( x , x )) ≥ u ( s k ( x , x )). • For x > x . By Claim A.11 we have that u ( s k ( x , x )) ≥ u ( x − , x ). Now, by mono-tonicity we have that u ( x − , x ) ≥ u ( x , x ) since because x and x are integers we havethat x − ≥ x . Then we have that u ( x , x ) ≥ u ( x , x ) which completes the proof.The two following claims allow us to show that the utility of the higher player is always greaterby relating between the utilities of player 1 and player 2. Claim A.10
If for every integers y ′ , z ′ and < k ′ ≤ k − s k ′ ( y ′ , z ′ ) is a subgame perfect equilibriumthen: u ( s k ( y, z )) ≥ u ( s k ( z, y )) for every y and z . Proof:
We prove the claim by induction on k the number of rounds. For the base case k = 1,observe that if y = z then clearly u ( s ( y, z )) ≥ u ( s ( z, y )); either because the players compete and c ( y, y ) ≥ (1 − c ( y, y )) or because the players do not compete and u ( s ( y, z )) = 1 > q = u ( s ( z, y )).Else, y = z , now if min { c ( y, z ) , (1 − c ( y, z )) } > q , then in both s ( y, z ) and s ( z, y ) the playerscompete and since y = z we have that c ( y, z ) = (1 − c ( y, z )), thus u ( s ( y, z )) = u ( s ( z, y )). Else,each of the players goes after a different candidate. If for example y > z , then in both games theplayer with reputation y goes after the stronger candidate and the player with reputation z goesafter the weaker candidate. Thus, u ( s k ( y, z )) = u ( s k ( z, y )).Next, we assume the correctness for ( k − k -round games. Wedistinguish between the following cases: 23. f ( s k ( y, z )) = f ( s k ( z, y )) = h + , + i : by using the induction hypothesis we get that that u ( s k − ( y + 1 , z )) ≥ u ( s k − ( z, y + 1)) and u ( s k − ( y, z + 1)) ≥ u ( s k − ( z + 1 , y )). Since c ( y, z ) ≥ (1 − c ( y, z )) this is sufficient for showing that u ( s k ( y, z )) ≥ u ( s k ( z, y )). Moregenerally this shows that u ( s h + , + i k ( y, z )) ≥ u ( s h + , + i k ( z, y )).2. f ( s k ( y, z )) = h + , + i and f ( s k ( z, y )) = h + , + i : by the definition of s k we have three possiblesubcases: • y < z : f ( s k ( y, z )) = h− , + i and f ( s k ( z, y )) = h + , −i . • y = z : f ( s k ( y, y )) = h + , −i . • y > z : f ( s k ( y, z )) = h + , −i and f ( s k ( z, y )) = h− , + i .It is not hard to see that for each one of these subcases we can use the induction hypothesisto show that the claim holds.3. f ( s k ( y, z )) = h + , + i and f ( s k ( z, y )) = h + , + i : if z > y , then: u ( s k ( z, y )) = 1 + u ( s k − ( z + 1 , y )) u ( s k ( y, z )) = (1 − c ( y, z )) · (1 + u ( s k − ( y, z + 1))) + c ( y, z ) · u ( s k − ( y + 1 , z ))By using monotonicity and applying the induction hypothesis we get that: u ( s k − ( y + 1 , z )) ≤ u ( s k − ( y, z + 1)) ≤ u ( s k − ( z + 1 , y ))Thus, the claim holds.Else, we have that y > z . We show that this case is not possible by a locking argument . Firstwe observe that this implies that player 1 is the lower player in the game G k ( z, y ) and in thefirst round of the game it prefers to go for the weaker candidate. Since we assume that forany y ′ , z ′ and 0 < k ′ ≤ k − s k ′ ( y ′ , z ′ ) is a subgame perfect equilibrium the requirementsof Corollary A.8 hold and thus we have that: u ( s k − ( z, y + 1)) = ( k − q . By applyingthe induction hypothesis we get that u ( s k − ( y + 1 , z )) ≤ u ( s k − ( z, y + 1)) = ( k − q .Now, since player 2 is the lower player in the game G k ( y + 1 , z ) we can apply Claim A.2 andconclude that u ( s k − ( y + 1 , z )) = ( k − q = u ( s k − ( z, y + 1)). Now, the following chain ofinequalities provides a contradiction for the assumption that in the game G k ( z, y ) the playersdo not compete as it shows that the lower player (player 1) actually prefers competing overgoing for the weaker candidate: u ( s h + , + i k ( z, y )) ≥ u ( s h + , + i k ( y, z )) > q + u ( s k ( y + 1 , z )) = q + u ( s k − ( z, y + 1)) = u ( s h− , + i k ( z, y )) . The claim is completed since we already treated the case in which y = z as part of the firsttwo cases.We now prove that a claim similar in spirit to the previous one also holds for player 2: Claim A.11
If for every integers y ′ , z ′ and < k ′ ≤ k − s k ′ ( y ′ , z ′ ) is a subgame perfect equilib-rium, then, u ( s k ( y, z + 1)) ≥ u ( s k ( z, y )) and u ( s k ( z, y )) ≥ u ( s k ( y, z + 1)) for every two integers y and z . Proof:
We prove the two inequalities by induction on k simultaneously. We begin with the basecase k = 1 and distinguish between the following cases:24. s ( y, z + 1) = s ( z, y ) = h + , + i : in this case u ( s ( y, z + 1)) = (1 − c ( y, z + 1)), u ( s ( z, y )) = c ( z, y ), u ( s ( z, y )) = (1 − c ( z, y )) and u ( s ( y, z + 1)) = c ( y, z + 1), thus the claim holds.2. s ( y, z + 1) = h + , + i and s ( z, y ) = h + , + i : if y ≤ z then u ( s ( y, z + 1)) = u ( s ( z, y )) =1 and u ( s ( y, z + 1)) = u ( s ( z, y )) = q , thus the claim holds. Else, y ≥ z + 1, then u ( s ( y, z + 1)) = u ( s ( z, y )) = q and u ( s ( y, z + 1)) = u ( s ( z, y )) = 1.3. The players compete in one of s ( y, z + 1) , s ( z, y ) and do not compete in the other: Observethat the following hold: • If y ≤ z then s ( y, z + 1) = h + , + i = ⇒ s ( z, y ) = h + , + i . Observe that (1 − c ( z, y )) ≥ c ( y, z + 1). This implies that u ( s h + , + i ( z, y )) ≥ u ( s h + , + i ( y, z + 1)) > q . Now since inboth cases the lower player is the player with reputation y the claim follows. • If y ≥ z + 1 then s ( z, y ) = h + , + i = ⇒ s ( y, z + 1) = h + , + i . Observe that (1 − c ( y, z +1)) ≥ c ( z, y ). This implies that u ( s h + , + i ( y, z + 1)) ≥ u ( s h + , + i ( z, y )) > q . Now since inboth cases the lower player is the player with reputation z or z + 1 the claim follows.Thus, we are left with the following two sub-cases:(a) s ( y, z +1) = h + , + i and s ( z, y ) = h− , + i : in this case: u ( s ( y, z +1)) = (1 − c ( y, z +1)), u ( s ( z, y )) = q , u ( s ( z, y )) = 1 and u ( s ( y, z + 1)) = c ( y, z + 1) and the claim holds.(b) s ( y, z +1) = h− , + i and s ( z, y ) = h + , + i : in this case: u ( s ( y, z +1)) = 1, u ( s ( z, y )) = c ( z, y ), u ( s ( z, y )) = (1 − c ( z, y )) and u ( s ( y, z + 1)) = q and the claim holds.Next, we assume correctness for ( k − k -round games. We distin-guish between the same cases as we did for the base case:1. f ( s k ( y, z + 1)) = f ( s k ( z, y )) = h + , + i : the players’ utilities are: u ( s k ( y, z + 1)) = (1 − c ( y, z + 1)) · (1 + u ( s k − ( y, z + 2))) + c ( y, z + 1) · u ( s k − ( y + 1 , z + 1)) u ( s k ( z, y )) = c ( z, y ) · (1 + u ( s k − ( z + 1 , y ))) + (1 − c ( z, y )) · u ( s k − ( z, y + 1)) u ( s k ( z, y )) = (1 − c ( z, y )) · (1 + u ( s k − ( z, y + 1))) + c ( z, y ) · u ( s k − ( z + 1 , y )) u ( s k ( y, z + 1)) = c ( y, z + 1) · (1 + u ( s k − ( y + 1 , z + 1))) + (1 − c ( y, z + 1)) · u ( s k − ( y, z + 2))It is not hard to see that by applying the induction hypothesis plus using monotonicity andthe facts that (1 − c ( y, z + 1)) ≥ c ( z, y ) and (1 − c ( z, y )) ≥ c ( y, z + 1) the claim holds. By thiswe have actually shown that a stronger statement holds: u ( s h + , + i k ( y, z +1)) ≥ u ( s h + , + i k ( z, y ))and u ( s h + , + i k ( z, y )) ≥ u ( s h + , + i k ( y, z + 1)).2. f ( s k ( y, z + 1)) = h + , + i and f ( s k ( z, y )) = h + , + i : if y ≤ z then u ( s k ( y, z + 1)) = 1 + u ( s k − ( y, z + 2)) ; u ( s k ( z, y )) = 1 + u ( s k − ( z + 1 , y )) u ( s k ( z, y )) = q + u ( s k − ( z + 1 , y )) ; u ( s k ( y, z + 1)) = q + u ( s k − ( y, z + 2)) . Thus we can use the induction hypothesis and get that the claim holds. Else, y ≥ z + 1, andwe can again write down the players’ utilities and apply the induction hypothesis to get thatthe claim holds.3. The players compete in one of f ( s k ( y, z + 1)) , f ( s k ( z, y )) and do not compete in the other:we will show that the following two lemmas hold:25 emma A.12 For y ≤ z , f ( s k ( y, z + 1)) = h + , + i = ⇒ f ( s k ( z, y )) = h + , + i . Proof:
We prove this by using a locking argument very similar to the one we used for ClaimA.10. Observe that in both games the lower player is the player with reputation y . Assumetowards a contradiction that in f ( s k ( z, y )) player 2 does not compete. Since player 2 is thelower player, we can use Corollary A.8 to get that u ( s k − ( z + 1 , y )) = ( k − q . By applyingthe induction hypothesis we get that ( k − q = u ( s k − ( z + 1 , y )) ≥ u ( s k − ( y, z + 2)). Now,since player 1 is the lower player in the game G k − ( y, z + 2) we can apply Claim A.2 andconclude that u ( s k − ( y, z + 2)) = ( k − q = u ( s k − ( z + 1 , y )). The following chain ofinequalities provides a contradiction that in the game G k ( z, y ) player 2 prefers to go for theweaker candidate over competing: u ( s h + , + i k ( z, y )) ≥ u ( s h + , + i k ( y, z + 1)) > q + u ( s k ( y, z + 2)) = q + u ( s k − ( z + 1 , y )) . Lemma A.13
For y ≥ z + 1 , f ( s k ( z, y )) = h + , + i = ⇒ f ( s k ( y, z + 1)) = h + , + i . Proof:
The proof is very similar to the previous lemma. Observe that in both gamesthe lower player is the player with reputation z or z + 1. Assume towards a contradictionthat in f ( s k ( y, z + 1)) player 2 does not compete. Since player 2 is the lower player, we canuse Corollary A.8 to get that u ( s k − ( y + 1 , z + 1)) = ( k − q . By applying the inductionhypothesis we get that ( k − q = u ( s k − ( y + 1 , z + 1)) ≥ u ( s k − ( z, y + 1)). Now, sinceplayer 1 is the lower player in the game G k − ( z, y + 1) we can apply Claim A.2 and concludethat u ( s k − ( z, y + 1)) = ( k − q = u ( s k − ( y + 1 , z + 1)). The following chain of inequalitiesprovide a contradiction that in the game G k ( y, z + 1) player 2 prefers to go for the weakercandidate over competing: u ( s h + , + i k ( y, z + 1)) ≥ u ( s h + , + i k ( z, y )) > q + u ( s k ( z, y + 1)) = q + u ( s k − ( y + 1 , z + 1)) . Thus, we are left with the following two sub-cases:(a) f ( s k ( y, z + 1)) = h + , + i and f ( s k ( z, y )) = h− , + i : observe that by using the inductionhypothesis and monotonicity it is not hard to see that the claim holds since: u ( s k ( y, z + 1)) = u ( s h + , + i k ( y, z + 1) > u ( s h− , + i k ( y, z + 1) ≥ u ( s h + , −i k ( z, y ) = u ( s k ( z, y )) u ( s k ( z, y )) = u ( s h + , −i k ( z, y )) ≥ u ( s h + , + i k ( z, y )) ≥ u ( s h + , + i k ( y, z + 1)) = u ( s k ( y, z + 1))(b) f ( s k ( z, y )) = h + , + i and f ( s k ( y, z + 1)) = h− , + i : observe that by using the inductionhypothesis and monotonicity it is not hard to see that the claim holds since: u ( s k ( z, y )) = u ( s h + , + i k ( z, y )) > u ( s h + , −i k ( z, y )) ≥ u ( s h− , + i k ( y, z + 1)) = u ( s k ( y, z + 1)) u ( s k ( y, z + 1)) = u ( s h + , −i k ( y, z + 1) ≥ u ( s h + , + i k ( y, z + 1) ≥ u ( s h + , + i k ( z, y ) = u ( s k ( z, y ))26 .2 Some More Properties of the Canonical Equilibrium We first show that if u i ( s k ( x , x )) = kq then player i goes for the weaker candidate in f ( s k ( x , x )).This immediately implies that if u i ( s k ( x , x )) = kq then player i in s k ( x , x ) goes for the weakercandidate in every round. Claim A.14 (generalization of Claim 2.4)
The following two statements hold: • u ( s k ( x , x )) = kq = ⇒ f ( s k ( x , x )) = h− , + i . • u ( s k ( x , x )) = kq = ⇒ f ( s k ( x , x )) = h + , −i . Proof:
We prove the claim for player 1 but a similar proof also works for player 2. Assumetowards a contradiction that u ( s k ( x , x )) = kq but f ( s k ( x , x )) = h + , + i . By Proposition A.5we have that s k ( x , x ) is a subgame perfect equilibrium, thus, if player 1 prefers to compete it hasto be the case that u ( s h + , + i k ( x , x )) > u ( s h− , + i k ( x , x )). Observe that u ( s h− , + i k ( x , x )) ≥ kq asa player can always guarantee itself a utility of at least kq in equilibrium. This is in contradictionto the assumption that f ( s k ( x , x )) = h + , + i .Next, based on Claim A.7 we can show that if a player competes and wins then in the nextround it prefers competing over going for the weaker candidate. Claim A.15 (generalization of Claim 2.5) If f ( s k ( x , x )) = h + , + i then: • f ( s k − ( x + 1 , x )) ∈ {h + , + i , h + , −i}• f ( s k − ( x , x + 1)) ∈ {h + , + i , h− , + i} Proof:
We first show that f ( s k − ( x + 1 , x )) ∈ {h + , + i , h + , −i} . Assume towards contradictionthat f ( s k − ( x + 1 , x )) = h− , + i . First observe that if c ( x , x ) > q then c ( x + 1 , x ) > q thusplayer 1 maximizes its utility by competing in the next round as well. It also has to be the casethat x + 1 < x since otherwise as the higher player in the game G k ( x + 1 , x ) player 1 shouldgo for the stronger candidate. By Corollary A.8 we have that u ( s k − ( x + 1 , x )) = ( k − q . Bymonotonicity we have that u ( s k − ( x , x + 1)) ≤ u ( s k − ( x + 1 , x )) = ( k − q . Thus, we havethat u ( s k ( x , x )) ≤ c ( x , x ) + ( k − q ≤ kq . This implies by Claim A.14 that player 1 does notcompete in f ( s k ( x , x )) in contradiction to the assumption. The proof of the second statementregarding player 2 is very similar and hence omitted. B Proofs from Section 3
Claim B.1
For any k , x and t > / − r ) such that r ≤ x + 1 t + 1 , u ( s k ( t − x, x )) ≤ max { b q ( k, t, x ) , kq } Proof:
We actually prove a stronger claim, which is that for t > / − r ) , b q ( k, t, x ) ≥ k . Theutility of any player in a game of k rounds is at most k and hence this will be enough to completethe proof. To do this we need to show that 3 F r ( x, t ) ≥ t > / − r ) . By [14] we havethat the median of a binomial distribution is at distance of at most ln(2) from its mean. Thus, if x ≥ rt + ln(2) we are done, as in this case 3 F r ( x, t ) ≥ . Else, rt + r − ≤ x < rt + ln(2). This27mplies that x + 2 ≥ rt + ln(2), which in turn implies that F r ( x + 2 , t ) ≥ . Since F r ( x + 2 , t ) = F r ( x, t ) + f r ( x + 1 , t ) + f r ( x + 2 , t ), what left to show is that f r ( x + 1 , t ) + f r ( x + 2 , t ) < : f r ( x + i, t ) = (cid:18) tx + i (cid:19) r x + i (1 − r ) t − ( x + i ) ≤ (cid:0) x + it (cid:1) − ( x + i ) r x + i (1 − r ) t − x − i ≤ r − ( x + i ) r x + i (1 − r ) t − x − i = (1 − r ) t − x − i ≤ (1 − r ) t − The last transition is due to the fact that x < rt + ln(2), r ≤ / i ≤
2. Thus we have that f r ( x + 1 , t ) + f r ( x + 2 , t ) ≤ − r ) t − . To compute when 2(1 − r ) t − < we take a naturallogarithm and get that: ( t − · ln(1 − r ) < ln( ), hence it is not hard to see that the claim holdsfor t > / − r ) . Claim B.2
For any k , x and t > / − r ) such that r > x + 1 t + 1 , u ( s k ( t − x, x )) ≤ max { b q ( k, t, x ) , kq } Proof:
Recall that b q ( k, t, x ) = xt + ( k − q + 3 F r ( x, t ) · (cid:0) ( k − − q ) + 1 (cid:1) . We prove the claimby induction. We first observe that the claim holds for k = 1. Notice that by the assumption that x +1 t +1 < r ≤ we have that player 2 is the lower player. Thus, u ( s ( t − x, x )) ≤ max (cid:8) xt , q (cid:9) and theclaim holds. Next, we assume correctness for k − k . If u ( s k ( t − x, x )) = kq ,then the induction hypothesis holds and we are done. Otherwise, we have that u ( s k ( t − x, x )) > kq ,this immediately implies that f ( s k ( t − x, x )) = h + , + i .By Claim 2.5 this implies that u ( s k − ( t − x, x + 1)) > ( k − q . Hence, either by the inductionhypothesis (if r > x +2 t +2 ) or by Claim B.1 (if r ≤ x +2 t +2 ) we have that u ( s k − ( t − x, x + 1)) ≤ b q ( k − , t + 1 , x + 1). Since x +1 t +2 < r . We can also use the induction hypothesis to get that u ( s k − ( t − x + 1 , x )) ≤ max { b q ( k − , t + 1 , x ) , kq } .To show that u ( s k ( t − x, x )) ≤ max { b q ( k, t, x ) , kq } we now distinguish between two cases:1. u ( s k − ( t − x + 1 , x )) ≤ b q ( k − , t + 1 , x ): u ( s k ( t − x, x )) ≤ xt (1 + b q ( k − , t + 1 , x + 1)) + t − xt b q ( k − , t + 1 , x )= xt + xt · x + 1 t + 1 + t − xt · xt + 1 + ( k − q + 3 F r ( x, t + 1) · (cid:0) ( k − − q ) + 1 (cid:1) + 3 xt · f r ( x + 1 , t + 1) · (cid:0) ( k − − q ) + 1 (cid:1) ≤ (1) xt + ( k − q + 3 F r ( x, t ) · (cid:0) ( k − − q ) + 1 (cid:1) ≤ (2) xt + ( k − q + 3 F r ( x, t ) · (cid:0) ( k − − q ) + 1 (cid:1) = b q ( k, t, x )Transition (1) is obtained by applying Claim B.3 (below) and some rearranging. For transition(2) we use the fact that xt < x +1 t +1 < r ≤ q .2. u ( s k − ( t − x + 1 , x )) = ( k − q : u ( s k ( t − x, x )) ≤ xt (1 + b q ( k − , t + 1 , x + 1)) + t − xt ( k − q = xt + xt · x + 1 t + 1 + xt ( k − q + xt · F r ( x + 1 , t + 1) · (cid:0) ( k − − q ) + 1 (cid:1) + t − xt ( k − q = xt + xt · x + 1 t + 1 + t − xt q + ( k − q + xt · F r ( x + 1 , t + 1) · (cid:0) ( k − − q ) + 1 (cid:1) ≤ xt + ( k − q + xt · F r ( x + 1 , t + 1) · (cid:0) ( k − − q ) + 1 (cid:1) xt · x +1 t +1 < xt · q since by assumption we have that x +1 t +1 < r ≤ q .Notice that xt F r ( x + 1 , t + 1) < F r ( x, t + 1) + xt f r ( x + 1 , t + 1). Hence by applying Claim B.3(below) we get that xt F r ( x + 1 , t + 1) < F r ( x, t ) which completes the proof. Claim B.3 If x < t then, F q ( x, t + 1) + xt f q ( x + 1 , t + 1) ≤ F q ( x, t ) Proof: F r ( x, t + 1) = x X i =0 (cid:18) t + 1 i (cid:19) q i (1 − q ) t +1 − i = (1 − q ) x X i =0 t + 1 t + 1 − i (cid:18) ti (cid:19) q i (1 − q ) t − i = (1 − q ) x X i =0 (1 + it + 1 − i ) (cid:18) ti (cid:19) q i (1 − q ) t − i = (1 − q ) F q ( x, t ) + (1 − q ) x X i =1 it + 1 − i · t ! i !( t − i )! q i (1 − q ) t − i = (1 − q ) F q ( x, t ) + (1 − q ) x X i =1 t !( i − t + 1 − i )! q i (1 − q ) t − i = (1 − q ) F q ( x, t ) + (1 − q ) x X i =1 (cid:18) ti − (cid:19) q i (1 − q ) t − i = (1 − q ) F q ( x, t ) + q x X i =1 (cid:18) ti − (cid:19) q i − (1 − q ) t − i +1 = (1 − q ) F q ( x, t ) + q x − X i =0 (cid:18) ti (cid:19) q i (1 − q ) t − i = (1 − q ) F q ( x, t ) + q · F q ( x − , t )= F q ( x, t ) − q · f q ( x, t )It remains to show that qf q ( x, t ) > xt qf q ( x + 1 , t + 1) which is done by noticing that since x < t then: xt f q ( x + 1 , t + 1) = xt (cid:18) t + 1 x + 1 (cid:19) q x +1 (1 − q ) t − x = xt · t + 1 x + 1 · q · (cid:18) tx (cid:19) q x (1 − q ) t − x = xt · t + 1 x + 1 · q · f q ( x, t ) < q · f q ( x, t )29 Other Competition Functions: Fixed Probability
One of the key components of our model is the underlying competition function : when players ofreputation x and x respectively compete for the same candidate in a given round, the competitionfunction specifies the probability that the candidate selects each player, in terms of x and x . Inthis section we explore the effect that using other competition functions has on the performanceratio. An extreme example is when the higher player deterministically wins the competition (and ifboth players have the same reputation, then each wins with probability 1 / G k ( x, x ) the players only compete for the first round to “discover” who is the higherplayer and then stop competing. Thus the performance ratio of this game is very close to 1. Inthis section we study a natural generalization of this function.Consider a competition function specifying that the lower player wins with a fixed probability p < /
2, and the higher player wins with probability (1 − p ). In case the two players have thesame reputations, ties are broken in favor of player 1. Clearly if p > q , then the players competeforever, since the lower player gains more from competing than from going for the weaker candidate.Therefore, we assume from now on that p < q . We observe that this competition function belongsto the set of competition functions defined in Section A of the appendix and hence we can make useof all claims specified there. For example, we have that the strategies s k ( x , x ) form a subgameperfect Nash equilibrium in this game (Proposition A.6 ), the players’ utilities are monotone (ClaimA.3) and that once a player decides to go after the lower candidate it will do so in all subsequentrounds (Claim A.7).We first show that once the absolute value of the difference between the players’ reputationsreaches θ (log( k )), the lower player stops competing (Claim C.1 and Lemma C.2). Then, in LemmaC.3 we show that the expected number of rounds it takes the players to reach such a differencein reputations, starting from equal reputations, is also θ (log( k )). This implies that as k goes toinfinity the performance ratio goes to 1, as we prove in Theorem C.4.Our first step is similar in spirit to the proof for the Tullock competition function; we show thatthe utility of the lower player is bounded by max { b pq ( k, d ) , kq } , where b pq ( k, d ) = p + ( p − p ) d k + (1 − ( p − p ) d )( k − q = p + ( p − p ) d (cid:0) ( k − − q ) + 1 (cid:1) + ( k − q This is obtained by induction over the difference in the reputations of the two players. Theintuition for the upper bound function is also similar. In the good event, the lower player becomesthe higher player and wins all subsequent rounds; hence its utility is k . In the bad event thelower player stays the lower player and loses the reward for competing this round; hence its utilityis at most ( k − q . To compute the probability of the good event, we can imagine that d (thedifference between the players’ reputations) is the initial location of a particle performing a biasedrandom walk that goes left with probability p and right with probability 1 − p . Under this view,the probability that this particle ever reaches 0 — and hence that the difference d ever reaches 0— is p − p [10]. We formalize this intuition in the next claim. The claim is stated and proved forplayer 2 but a similar claim also holds for player 1. Claim C.1
For any d ≥ and any k : u ( s k ( x, x − d )) ≤ max { b pq ( k, d ) , kq } . Proof:
First observe that the claim clearly holds for any k and d = 0, since for this case b pq ( k, d ) ≥ k and by definition we have that u ( s k ( x, x )) ≤ k . We now prove by induction on thenumber of rounds k that the claim holds for d ≥
1. Note that the claim holds for the base case,30 = 1, since u ( s ( x , x − d )) ≤ max { p, q } for every d ≥
1. We assume the claim holds for any0 < k ′ ≤ k − k . If u ( s k ( x, x − d )) = kq we are done. Else, u ( s k ( x, x − d )) = p (1 + u ( s k − ( x, x − d + 1))) + (1 − p )( u ( s k − ( x + 1 , x − d )))By the assumption that u ( s k ( x, x − d )) > kq and using Claim A.15 we have that u ( s k − ( x, x − d + 1)) > ( k − q . Thus, by the induction hypothesis (or our observation for d = 0 in case d was1), we have that u ( s k − ( x, x − d + 1)) ≤ b pq ( k − , d − u ( s k − ( x + 1 , x − d )) ≤ max { b pq ( k − , d + 1) , ( k − q } . We distinguish between two casesdepending on the two possible upper bounds on u ( s k − ( x + 1 , x − d )):If u ( s k − ( x + 1 , x − d )) ≤ b pq ( k − , d + 1): u ( s k ( x, x − d )) ≤ p (cid:0) b pq ( k − , d − (cid:1) + (1 − p ) · b pq ( k − , d + 1)= 2 p + ( k − q + ( p − p ) d − · (cid:18) p + p − p (cid:19) · (cid:0) ( k − − q ) + 1 (cid:1) = 2 p + ( p − p ) d (cid:0) ( k − − q ) + 1 (cid:1) + ( k − q< p + ( p − p ) d (cid:0) ( k − − q ) + 1 (cid:1) + ( k − q = b pq ( k, d )Else, u ( s k − ( x + 1 , x − d )) = ( k − q : u ( s k ( x, x − d )) ≤ p (cid:0) b pq ( k − , d − (cid:1) + (1 − p )( k − q< p + p − pq + ( k − q + p ( p − p ) d − (cid:0) ( k − − q ) + 1 (cid:1) < p + ( p − p ) d (cid:0) ( k − − q ) + 1 (cid:1) + ( k − q = b pq ( k, d )We can now use the previous claim to identify the magnitude of the difference between theplayers’ reputations for which they stop competing. We state the claim for player 2 but a similarclaim also holds for player 1. Lemma C.2 If d > log( k ) − log( p − q )log( − pp ) = d pq ( k ) then the lower player in the games G k ( x, x − d ) , G k ( x − d, x ) does not compete. Proof:
Claim C.1 reduces the problem of finding when does the lower player quits to solving for d such that b pq ( k, d ) < kq . This would be enough to conclude that player 2 stops competing as weknow by Claim A.14 that once u ( s k ( x , x )) = kq player 2 does not compete at all. After somerearranging of b pq ( k, d ) < kq we have that: (cid:0) p − p ) d (cid:0) ( k − − q ) + 1 (cid:1) < q − p We now take logarithms and get that: d · log( p − p ) + log(( k − − q ) + 1) < log( q − p )Therefore, d > log(( k − − q ) + 1)) − log( p − q )log( − pp ) and the claim follows.Next, we compute for how long the players are expected to compete until the absolute value ofthe difference between their reputations becomes greater than the previously computed bound. Todo this, we study this difference as it performs a biased random with a reflecting barrier at 0:31 emma C.3 In the game G k ( x , x ) , the expected number of rounds the players compete until theabsolute value of the difference between their reputations is at least d is at most d − p . Proof:
We consider a particle undergoing a biased random walk in which the probability ofmoving to the left is p and the probability of moving to the right is 1 − p , as before. Since thisparticle tracks the absolute value of the difference between the players’ reputations, the walk weare studying has a reflecting barrier on 0. This implies that when the particle reaches 0, in thenext step it always goes to 1.Our analysis will thus be based on studying the expected time it takes for the particle to reachthe value d , starting from a value below d . Clearly this expected time is maximized when theparticle starts at 0, corresponding to an initial reputation difference of 0. Thus, we compute abound on the expected number of rounds it takes players with identical reputations to reach adifference of d in their reputations.The expected time it takes the particle to reach d starting at 0 when there is a reflective barrieris upper bounded by the expected time it takes it to reach d starting at 0 when there is no suchbarrier. To see why, we invoke a standard argument in which we imagine both walks being governedby the random flips of a coin with bias p , and we compare between the trajectory of the particle inthese two walks for the same random sequence of coin-flip outcomes. We note that if the particlereaches d in the walk without the barrier it has to be the case that it also reached d using a prefixof the same sequence of coin-flip outcomes in the walk with the barrier.Finally, we use the fact that the expected number of rounds required for a particle performingthis walk to reach d starting from 0, without a barrier at 0, is d − p ([10]). To see why this is thecase, let E i be the expected time for the walk to reach i . If E is well defined, then E i = i · E .Also, E = 1 + (1 − p ) · p · E = 1 + 2 p · E . Therefore, E = 11 − p and E d = d − p .We are now ready to prove the following theorem. Theorem C.4
For fixed p < q the performance ratio of the game G k ( x , x ) is at least − θ ( log( k ) k ) . Proof:
Let R be a random variable equals to the number of rounds for which the players competein the game G k ( x , x ). We can use it to compute the social welfare as follows: u ( s k ( x , x )) = k X r =1 P r ( R = r )( k + ( k − r ) q ) = k (1 + q ) − q X r P r ( R = r ) · r Now, to compute a lower bound on the social welfare, we should compute an upper bound on P kr =1 P r ( R = r ) · r . We claim that P kr =1 P r ( R = r ) · r < d pq ( k )1 − p . The reason is that either thelower player quits when the difference between the players’ reputation is d pq ( k ), as we proved inClaim C.1, or the lower player might decide to quit earlier in the game. In any case, the expectednumber of rounds the players compete until the lower player drops is at most d pq ( k )1 − p , by LemmaC.3. Therefore, the performance ratio is at least k (1 + q ) − d pq ( k )1 − p qk (1 + q ) = 1 − θ ( log( kk
Let R be a random variable equals to the number of rounds for which the players competein the game G k ( x , x ). We can use it to compute the social welfare as follows: u ( s k ( x , x )) = k X r =1 P r ( R = r )( k + ( k − r ) q ) = k (1 + q ) − q X r P r ( R = r ) · r Now, to compute a lower bound on the social welfare, we should compute an upper bound on P kr =1 P r ( R = r ) · r . We claim that P kr =1 P r ( R = r ) · r < d pq ( k )1 − p . The reason is that either thelower player quits when the difference between the players’ reputation is d pq ( k ), as we proved inClaim C.1, or the lower player might decide to quit earlier in the game. In any case, the expectednumber of rounds the players compete until the lower player drops is at most d pq ( k )1 − p , by LemmaC.3. Therefore, the performance ratio is at least k (1 + q ) − d pq ( k )1 − p qk (1 + q ) = 1 − θ ( log( kk ) kk